Editing Smooth structure (section)

== Definition ==

A smooth structure on a manifold <math>M</math> is a collection of smoothly equivalent smooth atlases. Here, a '''smooth atlas''' for a topological manifold <math>M</math> is an [[Atlas (topology)|atlas]] for <math>M</math> such that each [[Transition map|transition function]] is a [[smooth map]], and two smooth atlases for <math>M</math> are '''smoothly equivalent''' provided their [[Union (set theory)|union]] is again a smooth atlas for <math>M.</math> This gives a natural [[equivalence relation]] on the set of smooth atlases.

A [[smooth manifold]] is a topological manifold <math>M</math> together with a smooth structure on <math>M.</math>

=== Maximal smooth atlases ===

By taking the union of all [[Atlas (topology)|atlases]] belonging to a smooth structure, we obtain a '''maximal smooth atlas'''. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases.   
Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.

In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. 
For example, if the manifold is [[Compact space|compact]], then one can find an atlas with only finitely many charts.

=== Equivalence of smooth structures ===

If <math>\mu</math> and <math>\nu</math> are two maximal atlases on <math>M</math> the two smooth structures associated to <math>\mu</math> and <math>\nu</math> are said to be equivalent if there is a [[diffeomorphism]] <math>f : M \to M</math> such that <math>\mu \circ f = \nu.</math> {{Citation needed|reason=It's not clear what composing a function with an atlas should mean here. |date=June 2020}}